a_{j}

contains individuals still under observation and event-free immediately before

a_{j}

. For group

k

its size is

r_{j}^{(k)} = \sum_{i = 1}^{n} 1 {g_{i} = k,, x_{i} \geq a_{j}} .

(1)

The use of

\geq

keeps the individual who experiences the event at

a_{j}

in the risk set just before that event.

ii

 0  0

Solution

 0  0

Under the null hypothesis of equal event-time distributions, every member of the combined risk set has the same instantaneous chance of being the next event. Conditional on one event at

a_{j}

and on the two risk-set sizes,

U_{j} \sim Bernoulli (\frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}}),

(1)

E_{0} (U_{j} ∣ r_{j}^{(0)}, r_{j}^{(1)}) = \frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}} .

(2)

iii

 0  0

Solution

 0  0

The quantity

u_{j} - E_{0} U_{j} = u_{j} - \frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}}

(1)

is the observed-minus-expected group-1 event count at time

a_{j}

. A positive value is local evidence that group 1 has the greater hazard function; a negative value points toward group 0.

iv

 0  0

Solution

 0  0

Summing the observed-minus-expected contributions gives the unstandardized log-rank statistic

T_{0} = \sum_{j = 1}^{m} {u_{j} - \frac{r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}}} .

(1)

Under the null it is centered at zero. A two-sided Log-rank test compares its magnitude with the square root of its null variance.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / a

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